How Do You Solve the 1D Heat Equation with Trigonometric Initial Conditions?

[prolix]

problem

u_t=u_xx, x is in [0,1], t>0

with
u(0,t)=u(1,t)=0, t>0
u(x,0)=sin(pi*x)-sin(3*pi*x), x is in (0,1)


i think its solution is of the form

u(x,t)=sigma(n=1 to infinity){a_n*sin(n*pi*x)*exp(-n^2*pi^2*t)

where a_n=2*integral(0 to 1){ (sin(pi*x)-sin(3*pi*x)) * sin(n*pi*x) }

but i have a_n = 0, for all n..

i don't know where is my mistake..

 

[the_wolfman]

Can you write your initial condition as a sin series?

$u(x,0)=\sum b_n sin(n \pi x)$

How does this compare to your sin series for

$u(x,t)$

Can you relate $a_n$ to $b_n$?